
We wish to come up with an efficient solver for (REF ). This is an unconstrained nonsmooth convex optimization, where the objective function is not differentiable due to the presence of the \(\ell _1\)  norm. However, since the proximal map of the weighted \(\ell _1\)  norm has a closed-form solution, this can be solved using a proximal algorithm such as ADMM [1]}, [2]}. The advantage of ADMM is that the original variable can be split into latent variables so that the ensuing subproblems can be solved efficiently, often in closed-form.
For example, a straightforward splitting for (REF ) results in the following equivalent problem:
\({\begin{array}{c}\min \ \frac{1}{2}\Vert \mathbf {Y}_\ell - \mathbf {S}\mathbf {P}_1\mathbf {E}\Vert ^2 + \frac{\lambda _1}{2}{\Vert \mathbf {Y}_h - \mathbf {P}_2\mathbf {E}\mathbf {R}\Vert }^2 \! + \!\frac{\lambda _2}{2} g(\mathbf {Q}) \\\mbox{s.t.} \quad \mathbf {P}_1=\mathbf {B}\mathbf {X}, \quad \mathbf {P}_2=\mathbf {X}, \\{P}_{i\tau }(\mathbf {X}) = \mathbf {Q}_{i\tau } \quad (i\in \Omega , \tau \in W),\end{array}}\) 
